## Rules for Maximizing Utility

### 1. Introduction

#### LEARNING OBJECTIVES

- Explain why maximizing utility requires that the last unit of each item purchased must have the same marginal utility per dollar
- Calculate the utility-maximizing choice

The problem of finding consumer equilibrium, that is, the combination of goods and services that will maximize an individual's total utility, comes down to comparing the trade-offs between one affordable combination (shown by a point on the budget line in Figure 1, below) with all the other affordable combinations.

Most people approach their utility-maximizing combination of choices in a step-by-step way. This step-by-step approach is based on looking at the tradeoffs, measured in terms of marginal utility, of consuming less of one good and more of another. You can think of this step-by-step approach as the "biggest bang for the buck" principle. For example, say that José starts off thinking about spending all his money on T-shirts and choosing point P, which corresponds to four T-shirts and no movies, as illustrated in Figure 1.

**Figure 1. A Choice between Consumption Goods.** José has income of $56. Movies cost $7 and T-shirts cost $14. The points on the budget constraint line show the combinations of movies and T-shirts that are affordable.

José chooses this starting point randomly; he has to start somewhere. Then he considers giving up the last T-shirt, the one that provides him the least marginal utility, and using the money he saves to buy two movies instead. Table 1 tracks the step-by-step series of decisions José needs to make (Key: T-shirts cost $14, movies cost $7, and Jose's income is $56).

Table 1. A Step-by-Step Approach to Maximizing Utility

Try | Which Has | Total Utility | Marginal Gain and Loss of Utility, Compared with Previous Choice | Conclusion |
---|---|---|---|---|

Choice 1: P | 4 T-shirts and 0 movies | 81 from 4 T-shirts + 0 from 0 movies = 81 | – | – |

Choice 2: Q | 3 T-shirts and 2 movies | 63 from 3 T-shirts + 31 from 0 movies = 94 | Loss of 18 from 1 less T-shirt, but gain of 31 from 2 more movies, for a net utility gain of 13 | Q is preferred over P |

Choice 3: R | 2 T-shirts and 4 movies | 43 from 2 T-shirts + 58 from 4 movies = 101 | Loss of 20 from 1 less T-shirt, but gain of 27 from two more movies for a net utility gain of 7 | R is preferred over Q |

Choice 4: S | 1 T-shirt and 6 movies | 22 from 1 T-shirt + 81 from 6 movies = 103 | Loss of 21 from 1 less T-shirt, but gain of 23 from two more movies, for a net utility gain of 2 | S is preferred over R |

Choice 5: T | 0 T-shirts and 8 movies | 0 from 0 T-shirts + 100 from 8 movies = 100 | Loss of 22 from 1 less T-shirt, but gain of 19 from two more movies, for a net utility loss of 3 | S is preferred over T |

Source: Clark Aldrich, https://courses.lumenlearning.com/wmopen-microeconomics/chapter/rules-for-maximizing-utility/

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